Problem: Let $p=x^2-2$. Which equation is equivalent to $(x^2-2)^2+18=9x^2-18$ in terms of $p$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $p^2-9p+18=0$ (Choice B) B $p^2-9p+36=0$ (Choice C) C $p^2+9p+18=0$ (Choice D) D $p^2+9p+36=0$
We are asked to rewrite the equation in terms of $p$, where ${p}={x^2-2}$. In order to do this, we need to find all of the places where the expression ${x^2-2}$ shows up in the equation, and then substitute ${p}$ wherever we see them! For instance, note that $9x^2-18=9({x^2-2})$. This means that we can rewrite the equation as: $(x^2-2)^2+18=9x^2-18$ $({x^2-2})^2+18=9({x^2-2})$ [What if I don't see this factorization?] Now we can substitute ${p}={x^2-2}$ : $({p})^2+18=9({p})$ Finally, let's manipulate this expression so that it shares the same form as the answer choices: ${p}^2-9{p}+18=0$ In conclusion, $p^2-9p+18=0$ is equivalent to the given equation when $p=x^2-2$.